In this section we define the entropic risk MAB problem. For the rest of the paper, denote \([k] = \lbrace 1,\dots ,k\rbrace \)  for any \(k \in \)  and \({(t)}^+ = \max \lbrace 0,t\rbrace \)  for \(t \in \) .

For any random variable \(X\) , given a risk parameter \(\gamma \) , the entropic risk [1]}, [2]} of \(X\)  is defined by
\(_\gamma (X) := \frac{1}{\gamma } \log [\exp (-\gamma X)].\) 
